21. Fractal Links - Amazing Seattle Fractals! tutorials or more information about fractals in general. Fractal TypesExplantions and illustrations of various types of fractals. http://www.fractalarts.com/ASF/Fractal_Links.html

Amazing Seattle Fractals! Home Fractal Art Galleries Fractal Tutorials Fractal Of The Week ... About Fractal Links I've included many resources on this page if you are looking for more information on fractals, including other fractal artists, tutorials or more information about fractals in general. If you are looking for fractal software programs check out my software page. Enjoy! Seattle Fractals Digital Art If you are interested in any of my art prints or downloading any of my screensavers for a free evaluation, you can find them here. High resolution art prints, fractal art galleries, fractal screensavers, custom made to order screensavers and more! Link Spectrum Fractal Tutorials and Related Links Fractal Types Explantions and illustrations of various types of fractals. UF Spiral Tutorial Dr. Joseph Trotsky's excellent tutorial on creating the classic fractal spiral form as well as other helpful UF info. He has also written helpful info on the program Fractal Explorer. Janet Parke Preslar's excellent tutorials on using the Ultra Fractal Program. Prof. John Matthew's

22. ! Echo Var= Title general Info on fractals Fractal Microscope (NCSA). fractals and Newton sMethod (Mathematica exercise). Nonlinear Dynamics FAQ (Jim Meiss) http://climate.gsfc.nasa.gov/~cahalan/FractalClouds/FractalGeneral.html

General Info on Fractals Chaos at Maryland (UMD-College Park) Fractal Compression (UCSD Institute for Nonlinear Science) Fractal Database (Spanky) Fractal FAQ (References, examples, computations, sofware,...) Fractal Microscope (NCSA) Fractals and Newton's Method (Mathematica exercise) Nonlinear Dynamics FAQ (Jim Meiss) Last Updated Site Maintained By: Dr. William Ridgway Responsible NASA Official: Dr. Robert Cahalan Privacy, Security, Notices NASA Goddard ... Atmospheres

23. What Is A Fractal? Fractal Is A Word Invented By Benoit Mandelbrot Before 3D quasifuchsian fractals, mainly three types of 3D fractals had been known . What are some general references on quasifuchsian fractals? http://www.fractal3d.com/faq/faq.html

What is a fractal? Fractal is a word invented by Benoit Mandelbrot to specify the complicated phenomena of shapes with self-similarity. According Benoit Mandelbrot words in his book, The Fractal Geometry of Nature, "I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "to break:" to create irregular fragents. It is therefore sensible - and how appropriate for our needs! - that, in addition to "fragmented" (as in fraction or refraction), fractus should also mean "irregular," both meanings being preserved in fragment." Other descriptions on fractals can be found from the following links: http://www.faqs.org/faqs/sci/fractals-faq/ sci.fractals FAQ (Q2: What is a fractal?) http://mathworld.wolfram.com/Fractal.html fractal: ERIC WEISSTEIN'S world of MATHEMATICS What is fractal3D? Fractal3D (www.fractal3D.com) is the world's first web site which provides information and products of great new finding 3D quasi-fuchsian fractals in pure mathematics. What is quasi-fuchsian fractals?

24. Graphics Archive - General Interest:Fractals:CCL (Science U) general InterestFractalsCCL. These images are pictures of two dimensional slicesof the cubic connectedness locus. The locus (CCL for short) is a http://www.scienceu.com/library/graphics/pix/General_Interest/Fractals/CCL/

Browse By: Graphics Archive Up Comments General Interest ... Fractals :CCL These images are pictures of two dimensional slices of the cubic connectedness locus. The locus (CCL for short) is a four-dimensional analog of the Mandelbrot set. Just like the Mandelbrot set parametrizes the dynamics of the quadratic family z->z^2+c, the CCL describes the cubic family z->z^3+az+b, where a and b are two complex parameters. The pictures were generated by the program Brot, developed at the Geometry Center by Apprentice Linus Upson and Summer Institute participant Christine Heitsch. Brot provides a friendly user interface and computational tool to explore varying families of two dimensional slices of objects such as the CCL. It was used in the research of Apprentice David Ben-Zvi into the geometry of hyperbolic components, which generalize the discs and cardioids making up the interior of the Mandelbrot set. "Sunrise on Io" and "Black Hole" made it into the October 1993 issue of Scientific American. Like the Mandelbrot set, the CCL is connected. (The images are typically not connected, since they only represent two dimensional slices which may connect in the other dimensions.) It is NOT however locally connected, as can be seen in "Sea of Mandelbrot", "Comb of Doom" and several others. Another interesting feature is the distinction between "complex" slices, which are reminiscent of the Mandelbrot set and its variants (see for example "Electric Storm", "Frosted Pane" or "Ships on the Sea"), and "real" slices, which look more like images of cantor sets and strange attractors arising in real dynamical systems. (A typical example is "Henon Lookalike".) Most pictures combine real and complex features in often overlapping configurations. (See "Cantor Whirlpool", "Jekyll and Hyde" and "Torrential Storm".)

25. Fractint Homepage Fractint now has a listserver set up for general questions and discussion aboutthe Fractint program and technical questions about fractals in general. http://spanky.triumf.ca/www/fractint/fractint.html

WELCOME to the Fractint WWW pages What is Fractint and what's it all about? Fractint is a freeware fractal generator created for IBMPC's and compatible computers. It is the most versatile and extensive fractal program available for any price. The authors (of which I am not one), work very hard to keep it that way. It has many great features and it is constantly being upgraded and improved by the Stone Soup team. Keep this link as a reference to stay up-to-date with the latest developments. Check out in the latest release. Check out in web resources. Check out on these pages. Getting Fractint Version 20.0 of Fractint Now available. New Image format on these Pages I've started the long process of replacing all the GIF images stored and displayed on these pages over to the PNG image type. If your web browser is having problems displaying them, it may mean you have to update/upgrade your browser. I apologise for any inconvenience, but the issue is being forced by the stance that Unisys has chosen in regard to LZW patent issues and the use of GIF images on Web servers. For more info on this topic see the Burn All Gifs site.

26. Spanky Other Fractal Docs His general page is titled Dynamical Systems and fractals and he has also madeavailable a series of lecture notes for a course on this topic. http://spanky.triumf.ca/www/other_docs.html

THE SPANKY FRACTAL DATABASE The picasso Fractal Fractal Information at Other Sites Check out the New Docs section of at Spanky page. Michael Barnsley's WWW site for Iterated Systems Inc. a commercial fractal compression company. Alexander Bogomolny alexb@cut-the-knot.com also has a page with information on the Feigenbaum phenomenom on a page titled Emergence of Chaos Dave Boll dboll@frii.com has in interesting page showing some of the relationships between Pi and the Mandelbrot set An "Introduction to Fractals from Paul Bourke in Auckland New Zealand. Mary Ann Connors mconnors@math.umass.edu from the Department of Mathematics and Statistics at the University of Massachusetts has created some web-documents describing self similar fractals. There is one titled Exploring Fractals and a modified version from PWS Kent publishers from their Math Modules Case Studies series. Robert L. Devany at Boston University presents some interesting pages on a variety of Fractal and related topics. The main page is titled: The Dynamical Systems and Technology Project and has many interesting links. Some of the most relavent here are: An introduction to

27. Bibliography General Fractals Bibliography general fractals. The original BibTeX file from which this filewas generated is available at http://www.dip.ee.uct.ac.za/~brendt/bibliographies/html/fractals.html

28. Grand Canyon Natural Fractals Presentation And Slide Notes I have linked to some general information about geology and fractals. It is goodfrom a perspective standpoint and fits in well with my presentation. http://astronomy.swin.edu.au/~pbourke/fractals/grandcanyon/

Natural Fractals in Grand Canyon National Park By Gayla Chandler Introduction I gave this presentation as a special program at Grand Canyon National Park in the Shrine of the Ages on Friday 31 December 2004 and Saturday 1 January 2005. The slideshow and notes are now accessible from the web for viewing by individuals and/or use by teachers in classrooms. What I ask in return is that no changes be made to the materials. Suggestions will be appreciated and changes made if deemed appropriate. Using the slides as slates for your own markings by highlighting similarity in the images may provide a richer viewing experience. (Instructions for doing this in PowerPoint are available on every slide.) Let this be an active experience. Play with it. The marks are only there while the slide is up. Every slide is shown below in thumbnail form with its respective notes. The thumbnails have very low resolution. The images in the presentation, however, have fairly good resolution, hopefully they will be projector quality (set the projector for 800x600 image size). This presentation in whole or part may not be used for profit in any way.

29. Fractals However, the most important fractals deviate from linear selfsimilarity. Some ofthese are fractals that describe general randomness, while others are http://www.fortunecity.com/emachines/e11/86/mandel.html

web hosting domain names photo sharing Fractals-a geometry of nature Fractal geometry plays two roles. It is the geometry of deterministic chaos and it can also describe the geometry of mountains, clouds and galaxies Benoit Mandelbrot Science and geometry have always progressed hand in hand. In the 17th century, Johannes Kepler found that he could represent the orbits of the planets around the Sun by ellipses. This stimulated Isaac Newton to explain these elliptical orbits as following from the law of gravity. Similarly, the back-and-forth motion of a perfect pendulum is represented by a sine wave. Simple dynamics used to be associated with simple geometrical shapes. This kind of mathematical picture implies a smooth relationship between an object's form and the forces acting on it. In the examples of the planets and the pendulum, it also implies that the physics is deterministic, meaning that you can predict the future of these systems from their past. Two recent developments have deeply affected the relationship between geometry and physics, however. The first comes from the recognition that nature is full of something called deterministic chaos. There are many apparently simple physical systems in the Universe that obey deterministic laws but nevertheless behave unpredictably . A pendulum acting under two forces, for example.

30. Fractals In general, fractals arising in a chaotic dynamical system have a far more complexscaling relation, usually involving a range of scales that can depend on http://www.drchaos.net/drchaos/Book/node9.html

Next: References and Notes Up: Some Terminology: MapsFlows, Previous: Binary Arithmetic Fractals Nature abounds with intricate fragmented shapes and structures, including coastlines, clouds, lightning bolts, and snowflakes. In 1975 Benoit Mandelbrot coined the term fractal to describe such irregular shapes. The essential feature of a fractal is the existence of a similar structure at all length scales. That is, a fractal object has the property that a small part resembles a larger part, which in turn resembles the whole object. Technically, this property is called self-similarity and is theoretically described in terms of a scaling relation. Chaotic dynamical systems almost inevitably give rise to fractals. And fractal analysis is often useful in describing the geometric structure of a chaotic dynamical system. In particular, fractal objects can be assigned one or more fractal dimensions, which are often fractional ; that is, they are not integer dimensions To see how this works, consider a Cantor set , which is defined recursively as follows (Fig.

31. Mathematics In Snowflake's Fractals Based on (and named after) Koch s famous snowflake curve , fractals like theones drawn by Snowflake are a classic example of fractals in general. http://compute2.shodor.org/snowflake/help_docs/sf_math.html

Mathematics in Snowflake's Fractals The fractals drawn by Snowflake have all kinds of interesting mathematics associated with them. (If you already know how the pictures are drawn , the following will make a lot more sense...) The quintessential fractal Based on (and named after) Koch's famous "snowflake curve", fractals like the ones drawn by Snowflake are a classic example of fractals in general. The concept of iterating a simple rule, and considering the infinite limit of that iterative process, is at the core of most, if not all, fractals in mathematics. Several of the concepts that characterize fractals are easy to see examples of and discuss with Snowflake. Self Similarity Self similarity is loosely considered the unifying quality of all things fractal. The curves explored by Snowflake are exactly self similar (that is, Snowflake draws curves that are approximations to exactly self similar curves). Essentially, that means that if you magnify any particular piece of the curve, it would look exactly like the original. (Some fractals are self similar in less exact waysfor example, the Mandelbrot set exhibits "quasi-self similarity". Magnified bits are like the orginal set, but with more intricacies. Some coastlines exhibit self similarity in that they show the same (non-Euclidean) dimension at different scales.) Non-Euclidean dimension One of the mind-boggling things about fractals is that they challenge our traditional notion of dimension. Fractal structures often fit best between the integer dimensions that we are used to from Euclidean geometry. For example, a common measure of dimension for the classical Koch snowflake curve is (log 4)/(log 3), approximately 1.26. With Snowflake, students can create their own fractals and learn how to find the fractal dimensions of these curves.

32. Question Corner -- Fractals And Their History Some of the modern interest in fractals among the general public comes from thecomputerassisted work of the IBM fractal project and 20th century http://www.math.toronto.edu/mathnet/questionCorner/fracthist.html

Navigation Panel: (These buttons explained below Question Corner and Discussion Area Fractals and their History Asked by Anuradha (last name unknown) on July 2, 1997 Please assist me in learning about "fractals." What are fractals? How are they applied in mathematics at secondary school levels and later at higher levels of school or education? The name "fractal" arises from the concept of a fractional dimension. What this means exactly is a difficult to say in simple language and we will simply try to give a feel for what fractals are and what sorts of behaviors they exhibit. Generally fractals are the result of some which is procedure repeated again and again. One of the most basic examples of a fractal is obtained in the following way. Start off with an equilateral triangle which has sides of length 1. Now on each edge of the triangle, add a new equilateral triangle with sides of length 1/3. Now in the middle of each side of this new shape, add a triangle with sides of length 1/9. Continue this process, each time adding new triangles to each side which are 1/3 the size of the triangles added in the last stage. When you are done (infinitely many steps later!) you have the desired fractal. Now let us take a moment to examine some of the properties of this strange new "shape." The area of this object can be calculated by adding up the area which we added at each stage. It is not difficult to see that this sum is actually a geometric series which converges to some finite area. One can also check that, after adding the new triangles at some stage, the perimeter of the shape is four-thirds what it used to be. Thus after repeating the process

33. DataCompression.info - Fractals This FAQ is not intended as a general introduction to fractals, or a set ofrigorous definitions, but rather a useful summary of ideas, sources, http://www.datacompression.info/Fractal.shtml

Sponsored by Visicron Fractals Fractal image compression is a lossy compression technique. Compression is performed by locating self-similar sections of an image, then using a fractal algorithm to generate the sections. Fractals are a very interesting field of study, but this page won't attempt to cover everything there is to know about fractals. For the most part, we will only look at fractals as they relate to compression. To learn more about this fascinating subject, try looking into some of the resources pointed to by the Fractal FAQ Search compression newsgroups for references to this topic Please be sure to visit Friends of DataCompression.info! Badtz Maru will be your guide. Books Fractals Everywhere Rate by Michael F. Barnsley. A revised and updated textbook focusing on how fractal geometry can be used to model real objects in the physical world. DCL reader TJ says This is, in my experience, the best mathematics book I have ever seen If you are interested in buying this book, please use the link on this page. Your purchase will help to support this site. Fractal Image Compression : Theory and Application Rate by Yuval Fisher (Editor). Featuring a collection of articles by twelve experts in the field of fractal image compression, this book contains the complete details of how to encode and decode images, offering working codes that are usable in applications. Includes some of the latest results in this field..

34. Fractal Art FAQ - General Information 1. FAQ general information. Table of contents This FAQ will offer theoreticalinformation about fractals and some paths to follow to the math’s domain, http://www.fractalus.com/fractal-art-faq/faq01.html

Table of contents Where to get the FAQ? How to contribute to the FAQ? What is the scope of the FAQ? ... Who have contributed to this FAQ? Version: Updated: Coordinator: Jean-Pierre Louvet. Maintainers: Jean-Pierre Louvet, Juan Luis Martínez, Phil Jackson (for fractal music). Another one or two volunteers are hoped. alt.binaries.pictures.fractals, alt.fractals, sci.fractals http://fractals.iut.u-bordeaux1.fr/F-art-faq/ (main site, modified address) http://www.fractalus.com/fractal-art-faq/ http://www.fractovia.org/F-art-faq/ for subscription information). But because not everybody is a member of this list, any request or suggestion formulated in any of the newsgroups listed above (in 1b) will be taken into consideration. However, in that case, send a copy of the message to the coordinator of this FAQ at: F-art.FAQ@hse.iuta.u-bordeaux.fr graphic art , but there is fractal music as well. Cynthia Church Dennis C. De Mars Dirk Meyer Frederik Slijkerman Janet Parke Preslar Jean-Pierre Louvet Juan Luis Martínez Ken Musgrave Kerry Mitchell Paul Carlson Paul Martz Phil Jackson Phil Thompson Rich Thomson (text from the previous sci.fractals-FAQ)

35. Fractal Curves And Dimension This sets are known as the selfsimilar fractals and, because of that ease, theproperty of is often considered to be germane to fractals in general. http://www.cut-the-knot.org/do_you_know/dimension.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Fractal Curves and Dimension Fractals burst into the open in early 1970s. Their breathtaking beauty captivated many a layman and a professional alike. Striking fractal images can often be obtained with very elementary means. However, the definition of fractals is far from being trivial and depends on a formal definition of dimension. It takes a few chapters of an Advanced Analysis book to rigorously define a notion of dimension. The important thing is that the notion is not unique and even more importantly, for a given set, various definitions may lead to different numerical results. When the results differ the set is called fractal. Or in the words of Benoit Mandelbrot [Ref 2], the father of fractals: A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension. The topological dimension of a smooth curve is, as one would expect, one and that of a sphere is two which may seem very intuitive. However, the formal definition was only given in 1913 by the Dutch mathematician L. Brouwer (1881-1966). A (solid) cube has the topological dimension of three because in any decomposition of the cube into smaller bricks there always are points that belong to at least four (3+1) bricks.

36. Fractal Web Sites Excellent source of information on fractal art and fractals in general. If you re interested in fractals or computer graphics in general, http://www.fractaldomains.com/html/sites.html

Fractal Web Sites Home News Download Register ... Fractal Domains Gallery The fractal sites listed on this page will lead you to dazzling images and mind boggling information about chaos and fractals. Even if you didn't find my site too entertaining, do yourself a favor and check out some of the sites listed below! Galleries Fractal Information Other Sites Of Interest Galleries Infinite Fractal Loop Index I already have a link to the Infinite Fractal Loop on my gallery page, but I want to put an extra plug in for the graphical index that was recently added for the Infinite Fractal Loop. The fractal gallery sites that are members of the loop are all of high quality, and this page has a representative thumbnail for practically every site in the loop. Takes a while for this page to load, but it's worth it! This is the single best starting point I know of for exploring fractal galleries on the Web an unparalleled portal to the world of fractal images. Fractalus (Damien M. Jones) Some of these fractals are really spectacular. Various formulas and coloring schemes are used. This is one of the first sites I've seen (besides my own) where anti-aliasing is was regularly used even on the large images. (Anti-aliasing is common at most of the best online galleries now.) Gumbycat's Cyberhome Gumbycat is the nom de plume of Linda Allison. Her talent continues to grow and amaze. She started out using Fractint, getting incredible shading effects from Fractint's limited color palette. Later she switched to Ultrafractal (a big favorite among PC fractal enthusiasts.)

37. Fractal - Enpsychlopedia fractals are said to possess infinite detail, and they may actually have aselfsimilar as well as a listing of tutorials regarding fractals in general, http://psychcentral.com/psypsych/Fractal

home resource directory disorders quizzes ... support forums Advertisement ( Fractal Missing image Mandelpart2.jpg The Mandelbrot set , named after its discoverer, is a famous example of a fractal. Listen to this article (info) This audio file was created from the revision dated , and does not reflect subsequent edits to the article. ( audio help More spoken articles A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. The term fractal was coined in by Benoît Mandelbrot , from the Latin fractus or "broken". Before Mandelbrot coined his term, the common name for such structures (the Koch snowflake , for example) was monster curve Fractals of many kinds were originally studied as mathematical objects.

38. Fractal Frequently Asked Questions And Answers This FAQ is not intended as a general introduction to fractals, or a set of rigorous Q28a What are some general references on fractals and chaos? http://www.faqs.org/faqs/fractal-faq/

MultiPage Fractal Frequently Asked Questions and Answers There are reader questions on this topic! Help others by sharing your knowledge From: stepp@muvms6.mu.wvnet.edu (Ermel Stepp) Newsgroups: sci.fractals stepp@marshall.edu mail-server@rtfm.mit.edu with the message: send usenet/news.answers/fractal-faq The hypertext version of the Fractal FAQ has hyperlinks to sources on the World Wide Web. It can be accessed with a browser such as xmosaic at http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html . Also, the hypertext version is online for review and comment at: http://www.marshall.edu/~stepp/fractal-faq/faq.html . Please suggest other links to add to the Fractal FAQ. For your information, the World Wide Web FAQ is available via: The WWW: http://sunsite.unc.edu/boutell/faq/www_faq.html alt.binaries.pictures.fractals http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html Fractal Microscope http://is.dal.ca:3400/~adiggins/fractal/ Dalhousie University Fractal Gallery http://acat.anu.edu.au/contours.html

39. Critical Phenomena In Natural Sciences: Chaos, Fractals, Selforganization And Di Critical Phenomena in Natural Sciences Chaos, fractals, In general the book,which evolved from a graduate student course, favours these last two http://www.iop.org/EJ/abstract/0305-4470/37/40/B03

@import url(http://ej.iop.org/style/nu/EJ.css); Quick guide Site map Athens login IOP login: Password: Create account Alerts Contact us Journals Home ... This issue J. Phys. A: Math. Gen. BOOK REVIEW Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools D Sornette Heidelberg: Springer-Verlag (2004) 528pp EUR79.95, £61.50, US$99.00 (hardback) ISBN 3-540-40754-5 Since the discovery of the renormalization group theory in statistical physics, the realm of applications of the concepts of scale invariance and criticality has pervaded several fields of natural and social sciences. This is the leitmotiv of Didier Sornette's book, who in Critical Phenomena in Natural Sciences reviews three decades of developments and applications of the concepts of criticality, scale invariance and power law behaviour from statistical physics, to earthquake prediction, ruptures, plate tectonics, modelling biological and economic systems and so on. This strongly interdisciplinary book addresses students and researchers in disciplines where concepts of criticality and scale invariance are appropriate: mainly geology from which most of the examples are taken, but also engineering, biology, medicine, economics, etc. A good preparation in quantitative science is assumed but the presentation of statistical physics principles, tools and models is self-contained, so that little background in this field is needed. The book is written in a simple informal style encouraging intuitive comprehension rather than stressing formal derivations. Together with the discussion of the main conceptual results of the discipline, great effort is devoted to providing applied scientists with the tools of data analysis and modelling necessary to analyse, understand, make predictions and simulate systems undergoing complex collective behaviour.

40. Chaffey's Fractal Links On The Web website for alumni of Chaffey High School, as well as general links resourcefor schools. fractals are very interesting and mathematical occurrences. http://www.chaffey.org/fractals/

Chaffey High School's FRACTALS on the Web http://www.chaffey.org/fractals/ Last Updated February 28, 2003 This site should be renovated soon, please excuse the webmaster's lack of construction. Welcome to the original Chaffey High School's Fractal Links of the Web. These pages where started as a hobby/extra curricular activity back in 1993, and have grown to be a pretty large website for alumni of Chaffey High School, as well as general links resource for schools. Fractals are very interesting and mathematical occurrences. Chances are, if you have opened your eyes, you've seen Fractals. This site, Chaffey's Fractal Links, is meant as a resource for everybody to use as a tool in search of fractal information, programs, articles and other fractal related pages on the Internet. This page has been up since 1993 and is continuing to grow in content. More will be coming soon... CHS Home Fractals Home About Fractals Awards ... Theory

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